| L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 11-s − 4·17-s − 4·19-s − 4·21-s + 6·23-s − 4·27-s + 2·29-s − 8·31-s + 2·33-s − 4·37-s − 6·41-s + 6·43-s + 2·47-s − 3·49-s − 8·51-s − 12·53-s − 8·57-s − 4·59-s + 14·61-s − 2·63-s − 10·67-s + 12·69-s − 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 1.25·23-s − 0.769·27-s + 0.371·29-s − 1.43·31-s + 0.348·33-s − 0.657·37-s − 0.937·41-s + 0.914·43-s + 0.291·47-s − 3/7·49-s − 1.12·51-s − 1.64·53-s − 1.05·57-s − 0.520·59-s + 1.79·61-s − 0.251·63-s − 1.22·67-s + 1.44·69-s − 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.191871686050505723023772771752, −7.22844652283952877540616710847, −6.72548837346794712093746847989, −5.91079469741736939188176872846, −4.88770977827577808579170180890, −3.98331514047171746360978742031, −3.28810060334468870197516710928, −2.57127588045710357001575296894, −1.67005665302555494693104159770, 0,
1.67005665302555494693104159770, 2.57127588045710357001575296894, 3.28810060334468870197516710928, 3.98331514047171746360978742031, 4.88770977827577808579170180890, 5.91079469741736939188176872846, 6.72548837346794712093746847989, 7.22844652283952877540616710847, 8.191871686050505723023772771752
